1 FINITE POINT CONFIGURATIONS J´Anos Pach

1 FINITE POINT CONFIGURATIONS J´anos Pach

INTRODUCTION

The study of combinatorial properties of finite point configurations is a vast area of research in geometry, whose origins go back at least to the ancient Greeks. Since it includes virtually all problems starting with “consider a set of n points in space,” space limitations impose the necessity of making choices. As a result, we will restrict our attention to Euclidean spaces and will discuss problems that we find particularly important. The chapter is partitioned into incidence problems (Section 1.1), metric problems (Section 1.2), and coloring problems (Section 1.3).

1.1 INCIDENCE PROBLEMS

In this section we will be concerned mainly with the structure of incidences between a finite point configuration P and a set of finitely many lines (or, more generally, k- dimensional flats, spheres, etc.). Sometimes this set consists of all lines connecting the elements of P . The prototype of such a question was raised by Sylvester [Syl93] more than one hundred years ago: Is it true that for any configuration of finitely many points in the plane, not all on a line, there is a line passing through exactly two points? This problem was rediscovered by Erd˝os [Erd43], and affirmative answers to this question was were given by Gallai and others [Ste44]. Generalizations for circles and conic sections in place of lines were established by Motzkin [Mot51] and Wilson-Wiseman [WW88], respectively.

GLOSSARY

Incidence: A point of configuration P lies on an element of a given collection of lines (k-flats, spheres, etc.). Simple crossing: A point incident with exactly two elements of a given collection of lines or circles. Ordinary line: A line passing through exactly two elements of a given point configuration. Ordinary circle: A circle passing through exactly three elements of a given point configuration. Ordinary hyperplane: A (d 1)-dimensional flat passing through exactly d elements of a point configuration− in Euclidean d-space. Motzkin hyperplane: A hyperplane whose intersection with a given d-dimensional point configuration lies—with the exception of exactly one point—in a (d 2)-dimensional flat. − 3

Preliminary version (August 10, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017. 4 J. Pach

Family of pseudolines: A family of two-way unbounded Jordan curves, any two of which have exactly one point in common, which is a proper crossing. Family of pseudocircles: A family of closed Jordan curves, any two of which have at most two points in common, at which the two curves properly cross each other. Regular family of curves: A family Γ of curves in the xy-plane deﬁned in terms of D real parameters satisfying the following properties. There is an integer s such that (a) the dependence of the curves on x, y, and the parameters is algebraic of degree at most s; (b) no two distinct curves of Γ intersect in more than s points; (c) for any D points of the plane, there are at most s curves in Γ passing through all of them. Degrees of freedom: The smallest number D of real parameters deﬁning a regular family of curves. Spanning tree: A tree whose vertex set is a given set of points and whose edges are line segments. Spanning path: A spanning tree that is a polygonal path. Convex position: P forms the vertex set of a convex polygon or polytope. k-Set: A k-element subset of P that can be obtained by intersecting P with an open halfspace. Halving plane: A hyperplane with P /2 points of P on each side. ⌊| | ⌋

SYLVESTER-TYPE RESULTS

1. Gallai theorem (dual version): Any set of lines in the plane, not all of which pass through the same point, determines a simple crossing. This holds even for families of pseudolines [KR72]. 2. Pinchasi theorem: Any set of at least ﬁve pairwise crossing unit circles in the plane determines a simple crossing [Pin01]. Any suﬃciently large set of pairwise crossing pseudocircles in the plane, not all of which pass through the same pair of points, determines an intersection point incident to at most three pseudocircles [ANP+04].

3. Pach-Pinchasi theorem: Given n red and n blue points in the plane, not all on a line, there always exists a bichromatic line containing at most two points of each color [PP00]. Any finite set of red and blue points contains a monochromatic spanned line, but not always a monochromatic ordinary line [Cha70]. 4. Motzkin-Hansen theorem: For any finite set of points in Euclidean d-space, not all of which lie on a hyperplane, there exists a Motzkin hyperplane [Mot51, Han65]. We obtain as a corollary that n points in d-space, not all of which lie on a hyperplane, determine at least n distinct hyperplanes. (A hyperplane is determined by a point set P if its intersection with P is not contained in a (d 2)-flat.) Putting the points on two skew lines in 3-space shows that the existence− of an ordinary hyperplane cannot be guaranteed for d> 2.

If n > 8 is suﬃciently large, then any set of n noncocircular points in the n 1 plane determines at least −2 distinct circles, and this bound is best possible [Ell67]. The minimum number of ordinary circles determined by n noncocircular points is 1 n2 O(n). Here lines are not counted as circles [LMM+17]. 4 − 5. Csima-Sawyer theorem: Any set of n noncollinear points in the plane determines at least 6n/13 ordinary lines (n > 7). This bound is sharp for n = 13 and false for n = 7 [KM58, CS93]; see Figure 1.1.1. Green-Tao theorem (formerly Motzkin-Dirac conjecture): If n is suﬃciently large and even, then the number of ordinary lines is at least n/2, and if n n 1 is odd, then at least 3 −4 . These bounds cannot be improved [GT13]. In 3-space, any set of n noncoplanar⌊ ⌋ points determines at least 2n/5 Motzkin hyperplanes [Han80, GS84].

FIGURE 1.1.1 Extremal examples for the (dual) Csima-Sawyer theorem: (a) 13 lines (including the line at inﬁnity) determining only 6 simple points; (b) 7 lines determining only 3 simple points.

(a) (b)

6. Orchard problem [Syl67]: What is the maximum number of collinear triples determined by n points in the plane, no four on a line? There are several constructions showing that this number is at least n2/6 O(n), which is asymptotically best possible, cf. [BGS74, FP84]. (See Figure− 1.1.2.) Green- Tao theorem [GT13]: If n is suﬃciently large, then the precise value of the n(n 3) maximum is − . ⌊ 6 ⌋

FIGURE 1.1.2 L 12 points and 19 lines, each passing through exactly 3 points.

7. Dirac’s problem [Dir51]: Does there exist a constant c0 such that any set of n points in the plane, not all on a line, has an element incident to at least n/2 c connecting lines? If true, this result is best possible, as is − 0

shown by the example of n points distributed as evenly as possible on two intersecting lines. (It was believed that, apart from some small examples listed in [Grü72], this statement is true with c0 = 0, until Felsner exhibited an infinite series of configurations, showing that c0 3/2; see [BMP05, p. 313].) It is known [PW14] that there is a positive constant≥ c such that one can find a point incident to at least cn connecting lines. The best known value of the constant for which this holds is c =1/37. A useful equivalent formulation of this assertion is that any set of n points in the plane, no more than n k of − which are on the same line, determines at least c′kn distinct connecting lines, for a suitable constant c′ > 0. Note that according to the d = 2 special case of the Motzkin-Hansen theorem, due to Erd˝os (see No. 4 above), for k = 1 the number of distinct connecting lines is at least n. For k = 2, the corresponding bound is 2n 4, (n 10). − ≥ 8. Ungar’s theorem [Ung82]: n noncollinear points in the plane always determine at least 2 n/2 lines of different slopes (see Figure 1.1.3); this proves Scott’s conjecture.⌊ Furthermore,⌋ any set of n points in the plane, not all on a line, permits a spanning tree, all of whose n 1 edges have different slopes [Jam87]. Pach, Pinchasi, and Sharir [PPS07] proved− Scott’s conjecture in 3 dimensions: any set of n 6 points in R3, not all of which are on a plane, determine at least 2n 5 pairwise≥ nonparallel lines if n is odd, and at least 2n 7 if n is even. This− bound is tight for every odd n. −

FIGURE 1.1.3 7 points determining 6 distinct slopes.

UPPER BOUNDS ON THE NUMBER OF INCIDENCES

Given a set P of n points and a family Γ of m curves or surfaces, the number of incidences between them can be obtained by summing over all p P the number of elements of Γ passing through p. If the elements of Γ are taken∈ from a regular family of curves with D degrees of freedom, the maximum number of incidences D/(2D 1) (2D 2)/(2D 1) between P and Γ is O(n − m − − + n + m). In the most important applications, Γ is a family of straight lines or unit circles in the plane (D = 2), or it consists of circles of arbitrary radii (D = 3). The best upper bounds known for the number of incidences are summarized in Table 1.1.1. It follows from the ﬁrst line of the table that for any set P of n points in the plane, the number of distinct straight lines containing at least k elements of P is O(n2/k3 + n/k), and this bound cannot be improved (Szemer´edi-Trotter). In the second half of the table, κ(n,m) and β(n,m) denote extremely slowly growing functions, which are certainly o(nǫmǫ) for every ǫ > 0. A family of pseudocircles is special if its curves admit a 3-parameter algebraic representation. A collection of spheres in 3-space is said to be in general position here if no three of them pass through the same circle [CEG+90, ANP+04].

TABLE 1.1.1 Maximum number of incidences between n points of P and m elements of Γ. [SzT83, CEG+90, NPP+02, KMSS12, Z13]

PT. SET P FAMILY Γ BOUND TIGHT Planar lines O(n2/3m2/3 + n + m) yes Planar pseudolines O(n2/3m2/3 + n + m) yes Planar unit circles O(n2/3m2/3 + n + m) ? Planar pairwise crossing circles O(n1/2m5/6 + n2/3m2/3 + n + m) ? Planar special pseudocircles O(n6/11m9/11κ(n, m)+ n2/3m2/3 + n + m) ? Planar pairwise crossing pseudocircles O(n2/3m2/3 + n + m4/3) ? 3-dim’l unit spheres O(n3/4m3/4 + n + m) ? 3-dim’l spheres in gen. position O(n3/4m3/4 + n + m) ? d-dim’l circles O(n6/11m9/11κ(n, m)+ n2/3m2/3 + n + m) ?

MIXED PROBLEMS

Many problems about finite point configurations involve some notions that cannot be defined in terms of incidences: convex position, midpoint of a segment, etc. Below we list a few questions of this type. They are discussed in this part of the chapter, and not in Section 1.2, which deals with metric questions, because we can disregard most aspects of the Euclidean metrics in their formulation. For example, convex position can be defined by requiring that some sets should lie on one side of certain hyperplanes. This is essentially equivalent to introducing an order along each straight line. 1. Erd˝os-Klein-Szekeres problem: What is the maximum number of points that can be chosen in the plane so that no three are on a line and no k are in convex position (k> 3)? Denoting this number by c(k), it is known that

k 2 k+o(k) 2 − c(k) 2 . ≤ ≤ The lower bound given in [ES61] is conjectured to be tight, but this has been 2k 4 verified only for k 6; [SP06]. The original upper bound, k −2 in [ES35] ≤ 2k 5 2k 8 − was successively improved to − − +1, where the last bound is due k 2 − k 3 to Mojarrad and Vlachos [MV16]. − Suk − [Suk17] proved that, as k tends to infinity, the lower bound is asymptotically tight. Let e(k) denote the maximum size of a planar point set P that has no three elements on a line and no k elements that form the vertex set of an “empty” convex polygon, i.e., a convex k-gon whose interior is disjoint from P . We have e(3) = 2, e(4) = 4, e(5) = 8, e(6) < , and Horton showed that e(k) is infinite for all k 7 [Har78, Hor83, Nic07,∞ Ger08]. ≥ d 2. The number of empty k-gons: Let Hk (n) (n k d+1) denote the minimum number of k-tuples that induce an empty convex≥ ≥ polytope of k vertices in a set of n points in d-space, no d + 1 of which lie on a hyperplane. Clearly, H1(n)= n 1 and H1(n)=0 for k> 2. For k = d + 1, we have 2 − k 1 2 lim Hd(n)/nd , d! ≤ n k ≤ (d 1)! →∞ −

2 2 2 [Val95]. For d = 2, the best estimates known for Hk = limn Hk (n)/n are given in [BV04]: →∞

1 H2 1.62, 1/2 H2 1.94, 0 H2 1.02, ≤ 3 ≤ ≤ 4 ≤ ≤ 5 ≤ 0 H2 0.2, H2 = H2 = ... =0. ≤ 6 ≤ 7 8 d 3. The number of k-sets [ELSS73]: Let Nk (n) denote the maximum number of k-sets in a set of n points in d-space, no d + 1 of which lie on the same d hyperplane. In other words, Nk (n) is the maximum number of diﬀerent ways in which k points of an n-element set can be separated from the others by a hyperplane. It is known that

Ω(√log k) 2 1/3 ne Nk (n) O n(k + 1) , ≤ ≤ 3 5/2 [T´ot01, Dey98]. For the number of halving planes, N n/2 (n)= O(n ), and ⌊ ⌋ d 1 Ω(√log n) d d n − e ) N n/2 (n)= o(n ). ≤ ⌊ ⌋ n The most interesting case is k = 2 in the plane, which is the maximum number of distinct ways to cut a set of n points in the plane in half (number of halving lines).

FIGURE 1.1.4 12 points determining 15 combinatorially distinct halving lines.

The maximum number of at-most-k-element subsets of a set of n points in d/2 d/2 d-space, no d + 1 of which lie on a hyperplane, is O n⌊ ⌋k⌈ ⌉ , and this bound is asymptotically tight [CS89]. In the plane the maximum number n of at-most-k-element subsets of a set of n points is kn for k < 2 , which is reached for convex n-gons [AG86, Pec85]. 4. The number of midpoints: Let M(n) denote the minimum number of diﬀerent n midpoints of the 2 line segments determined by n points in convex position

in the plane. One might guess that M(n) (1 o(1)) n , but it was shown ≥ − 2 in [EFF91] that

1/2 2 n n(n + 1)(1 e− ) n n 2n + 12 − M(n) − . 2 − 4 ≤ ≤ 2 − 20

5. Midpoint-free subsets: As a partial answer to a question proposed in [BMP05], it was proved by V. B´alint et al.[BBG+95] that if m(n) denotes the largest number m such that every set of n points in the plane has a midpoint-free subset of size m, then

1+ √8n +1 − m(n). 2 ≤

1 ǫ However, asymptotically, Ω(n − )n m(n) o(n), for every ǫ> 0. ≤ ≤

OPEN PROBLEMS

Here we give six problems from the multitude of interesting questions that remain open. 1. Does there exist for every k a number n = n(k) such that in any set P of n points in the plane there are k elements that are either collinear and pairwise see each other? (Two elements of P see each other if the segment connecting them does not pass through any other point of P . [KPW05]. 2. Generalized orchard problem (Grünbaum): What is the maximum number ck(n) of collinear k-tuples determined by n points in the plane, no k +1 2 of which are on a line (k 3)? In particular, show that c4(n) = o(n ). Improving earlier lower bounds≥ of Grünbaum [Grü76] and Ismailescu [Ism02], Solymosi and Stojakovićshowed that

γ 2 k c (n) n − √log n , k ≥

for a suitable γk > 0. [SS13]. For k = 3, according to the Green-Tao theorem n(n 3) [GT13], we have c (n)= − , provided that n is sufficiently large. 3 ⌊ 6 ⌋ 3. Maximum independent subset problem (Erd˝os): Determine the largest number α(n) such that any set of n points in the plane, no four on a line, has an α(n)-element subset with no collinear triples. Füredi [Für91] has shown that Ω(√n log n) α(n) o(n). ≤ ≤ 4. Slope problem (Jamison): Is it true that every set of n points in the plane, not all on a line, permits a spanning path, all of whose n 1 edges have different slopes? − 5. Empty triangle problem (Bárány): Is it true that every set of n points in the plane, no three on a line, determines at least t(n) empty triangles that share a side, where t(n) is a suitable function tending to infinity?

6. Balanced partition problem (Kupitz): Does there exist an integer k with the property that for every planar point set P , there is a connecting line such that the diﬀerence between the number of elements of P on its left side and right side does not exceed k? Some examples due to Alon show that this assertion is not true with k = 1. Pinchasi proved that there is a connecting line, for which this diﬀerence is O(log log n).

1.2 METRIC PROBLEMS

n The systematic study of the distribution of the 2 distances determined by n points was initiated by Erd˝os in 1946 [Erd46]. Given a point conﬁguration P = p1,p2,...,pn , let g(P ) denote the number of distinct distances determined by P , and{ let f(P )} denote the number of times that the unit distance occurs between two elements of P . That is, f(P ) is the number of pairs pipj (i

GLOSSARY

Unit distance graph: A graph whose vertex set is a given point conﬁguration P , in which two points are connected by an edge if and only if their distance is one. Diameter: The maximum distance between two points of P . General position in the plane: No three points of P are on a line, and no four on a circle. Separated set: The distance between any two elements is at least one. Nearest neighbor of p P: A point q P , whose distance from p is minimum. ∈ ∈ Farthest neighbor of p P: A point q P , whose distance from p is maximum. ∈ ∈ Homothetic sets: Similar sets in parallel position.

REPEATED DISTANCES

Extremal graph theory has played an important role in this area. For example, it is easy to see that the unit distance graph assigned to an n-element planar point set P cannot contain K2,3, a complete bipartite graph with 2 and 3 vertices in its classes. Thus, by a well-known graph-theoretic result, f(P ), the number of edges in this graph, is at most O(n3/2). This bound can be improved to O(n4/3) by using more sophisticated combinatorial techniques (apply line 3 of Table 1.1.1 with m = n); but we are still far from knowing what the best upper bound is. In Table 1.2.1, we summarize the best currently known estimates on the maximum number of times the unit distance can occur among n points in the plane, under various restrictions on their position. In the ﬁrst line of the table—and throughout this chapter—c denotes (unrelated) positive constants. The second and

TABLE 1.2.1 Estimates for the maximum number of unit distances determined by an n-element planar point set P .

POINT SET P LOWER BOUND UPPER BOUND SOURCE Arbitrary n1+c/ log log n O(n4/3) [Erd46, SST84] Separated 3n √12n 3 3n √12n 3 [Reu72, Har74] ⌊ − − ⌋ ⌊ − − ⌋ Of diameter 1 n n [HP34] In convex position 2n 7 O(n log n) [EH90, Für90] − No 3 collinear Ω(n log n) O(n4/3) Kárteszi Separated, no 3 coll. (2 + 5/16 o(1))n (2 + 3/7)n [Tót97] −

third lines show how many times the minimum distance and the maximum distance, resp., can occur among n arbitrary points in the plane. Table 1.2.2 contains some analogous results in higher dimensions.

FIGURE 1.2.1 A separated point set with ⌊3n − (12n − 3)1/2⌋ unit distances (n = 69). All such sets have been characterized by Kupitz.

TABLE 1.2.2 Estimates for the maximum number of unit distances determined by an n-element point set P in d-space.

POINT SET P LOWER BOUND UPPER BOUND SOURCE d = 3, arbitrary Ω(n4/3 log log n) O(n3/2) [Erd60, KMS12, Zal13] d = 3, separated 6n O(n2/3) 6n Ω(n2/3) Newton − − d = 3, diameter 1 2n 2 2n 2 [Gr¨u56, Hep56] − − d = 3, on sphere Ω(n4/3) O(n4/3) [EHP89] (rad. 1/√2) 4/3 d = 3, on sphere Ω(n log∗ n) O(n ) [EHP89] (rad. r = 1/√2) 6 2 2 d = 4 n + n 1 n + n [Bra97, Wam99] ⌊ 4 ⌋ − ⌊ 4 ⌋ n2 1 n2 1 d 4 even, arb. 2 1 d/2 +n O(d) 2 1 d/2 +n Ω(d) [Erd67] ≥ −⌊ ⌋ − −⌊ ⌋ − n2 1 4/3 n2 1 4/3 d> 4 odd, arb. 2 1 d/2 +Ω(n ) 2 1 d/2 +O(n ) [EP90] −⌊ ⌋ −⌊ ⌋

The second line of Table 1.2.1 can be extended by showing that the smallest distance cannot occur more than 3n 2k +4 times between points of an n-element −

FIGURE 1.2.2 n points, among which the second- 24 smallest distance occurs ( 7 + o(1))n times.

set in the plane whose convex hull has k vertices [Bra92a]. The maximum number of occurrences of the second-smallest and second-largest distance is (24/7+ o(1))n and 3n/2 (if n is even), respectively [Bra92b, Ves78]. Given any point configuration P , let Φ(P ) denote the sum of the numbers of farthest neighbors for every element p P . Table 1.2.3 contains tight upper bounds on Φ(P ) in the plane and in 3-space,∈ and asymptotically tight ones for higher dimensions [ES89, Csi96, EP90]. Dumitrescu and Guha [DG04] raised the following related question: given a colored point set in the plane, its heterocolored diameter is the largest distance between two elements of different color. Let φk(n) denote the maximum number of times that the heterocolored diameter can occur in a k-colored n-element point set between two points of different color. It is known that φ2(n)= n, φ3(n) and φ4(n)=3n/2+ O(1) and φk(n)= O(n) for every k.

TABLE 1.2.3 Upper bounds on Φ(P ), the total number of farthest neighbors of all points of an n-element set P .

POINT SET P UPPER BOUND SOURCE Planar, n is even 3n 3 [ES89, Avi84] − Planar, n is odd 3n 4 [ES89, Avi84] − Planar, in convex position 2n [ES89] 3-dimensional, n 0 (mod 2) n2/4 + 3n/2 + 3 [Csi96, AEP88] ≡ 3-dimensional, n 1 (mod 4) n2/4 + 3n/2 + 9/4 [Csi96, AEP88] ≡ 3-dimensional, n 3 (mod 4) n2/4 + 3n/2 + 13/4 [Csi96, AEP88] ≡ d-dimensional (d> 3) n2(1 1/ d/2 + o(1)) − ⌊ ⌋

DISTINCT DISTANCES

It is obvious that if all distances between pairs of points of a d-dimensional set P are the same, then P d +1. If P determines at most g distinct distances, we have that P d+|g ;| see≤ [BBS83]. This implies that if d is ﬁxed and n tends to | | ≤ d inﬁnity, then the minimum number of distinct distances determined by n points in 1/d d-space is at least Ω(n ). Denoting this minimum by gd(n), for d 3 we have the following results: ≥

2/d 2/(d(d+2)) 2/d Ω(n − ) g (n) O(n ). ≤ d ≤ The lower bound is due to Solymosi and Vu [SV08], the upper bound to Erd˝os. In the plane, Guth and Katz [GK15] nearly proved Erd˝os’s conjecture by showing that g2(n)=Ω(n/ log n). Combining the results in [SV08] and [GK15], de Zeeuw showed that 3/5 g3(n)=Ω(n / log n). In Table 1.2.4, we list some lower and upper bounds on the minimum number of distinct distances determined by an n-element point set P , under various assumptions on its structure.

TABLE 1.2.4 Estimates for the minimum number of distinct distances determined by an n-element point set P in the plane.

POINT SET P LOWER BOUND UPPER BOUND SOURCE Arbitrary Ω(n/ log n) O(n/√log n) [GK15] In convex position n/2 n/2 [Alt63] ⌊ ⌋ ⌊ ⌋ No 3 collinear (n 1)/3 n/2 Szemer´edi [Erd75] ⌈ − ⌉ ⌊ ⌋ In general position Ω(n) O(n1+c/√log n) [EFPR93]

RELATED RESULTS

1. Integer distances: There are arbitrarily large, noncollinear finite point sets in the plane such that all distances determined by them are integers, but there exists no infinite set with this property [AE45]. 2. Generic subsets: Any set of n points in the plane contains Ω((n/ log n)1/3) points such that all distances between them are distinct [Cha13]. This bound could perhaps be improved to about n1/2/(log n)1/4, which would be best possible, as is shown by the example of the √n √n integer grid. The cor- ×1/(3d 3) 1/3 2/(3d 3) responding quantity in d-dimensional space is Ω(n − (log n) − − . [CFG+15]. 3. Borsuk’s problem: It was conjectured that every (finite) d-dimensional point set P can be partitioned into d + 1 parts of smaller diameter. It follows from the results quoted in the third lines of Tables 1.2.1 and 1.2.2 that this is true for d = 2 and 3. Surprisingly, Kahn and Kalai [KK93] proved that

there exist sets P that cannot be partitioned into fewer than (1.2)√d parts of smaller diameter. In particular, the conjecture is false for d 64 [JB14]. On the other hand, it is known that for large d, every d-dimensional≥ set can be partitioned into ( 3/2+ o(1))d parts of smaller diameter [Sch88]. p 4. Nearly equal distances: Two numbers are said to be nearly equal if their difference is at most one. If n is sufficiently large, then the maximum number of times that nearly the same distance occurs among n separated points in the plane is n2/4 . The maximum number of pairs in a separated set of n points in the⌊ plane,⌋ whose distance is nearly equal to any one of k arbitrarily 2 chosen numbers, is n (1 1 + o(1)), as n tends to infinity [EMP93]. 2 − k+1 5. Repeated angles: In an n-element planar point set, the maximum number of noncollinear triples that determine the same angle is O(n2 log n), and this bound is asymptotically tight for a dense set of angles (Pach-Sharir). The corresponding maximum in 3-space is at most O(n7/3). In 4-space the angle π/2 can occur Ω(n3) times, and all other angles can occur at most O(n5/2+ε) times, for every ε > 0 [Pur88, AS05]. For dimension d 5 all angles can occur Ω(n3) times. ≥

6. Repeated areas: Let td(n) denote the maximum number of triples in an n- element point set in d-space that induce a unit area triangle. It is known 2 20/9 17/7+ε that Ω(n log log n) t2(n) O(n ), t3(n) = O(n ), for every ε > ≤3 ≤ 3 0; t4(n),t5(n) = o(n ), and t6(n) = Θ(n ) ([EP71, PS90, RS17, DST09]). Maximum- and minimum-area triangles occur among n points in the plane at most n and at most Θ(n2) times [BRS01].

7. Congruent triangles: Let Td(n) denote the maximum number of triples in an n-element point set in d-space that induce a triangle congruent to a given triangle T . It is known [AS02, AFM02]´ that Ω(n1+c/ log log n) T (n) O(n4/3), ≤ 2 ≤ Ω(n4/3) T (n) O(n5/3+ǫ), ≤ 3 ≤ Ω(n2) T (n) O(n2+ǫ), ≤ 4 ≤ 7/3 T5(n)=Θ(n ), and T (n)=Θ(n3) for d 6. d ≥ 8. Similar triangles: There exists a positive constant c such that for any triangle T and any n 3, there is an n-element point set in the plane with at least cn2 triples that≥ induce triangles similar to T . For all quadrilaterals Q, whose points, as complex numbers, have an algebraic cross ratio, the maximum number of 4-tuples of an n-element set that induce quadrilaterals similar to Q is Θ(n2). For all other quadrilaterals Q, this function is slightly subquadratic. The maximum number of pairwise homothetic triples in a set of n points in the plane is O(n3/2), and this bound is asymptotically tight [EE94, LR97]. The number of similar tetrahedra among n points in three-dimensional space is at most O(n2.2) [ATT98]. Further variants were studied in [Bra02]. 9. Isosceles triangles, unit circles: In the plane, the maximum number of triples that determine an isosceles triangle, is O(n2.137) (Pach-Tardos). The maximum number of distinct unit circles passing through at least 3 elements of a planar point set of size n is at least Ω(n3/2) and at most n2/3 O(n) [Ele84]. −

CONJECTURES OF ERDOS˝

1. The number of times the unit distance can occur among n points in the plane does not exceed n1+c/ log log n.

2. Any set of n points in the plane determines at least Ω(n/√log n) distinct distances. 3. Any set of n points in convex position in the plane has a point from which there are at least n/2 distinct distances. ⌊ ⌋ 4. There is an integer k 4 such that any ﬁnite set in convex position in the plane has a point from≥ which there are no k points at the same distance.

5. Any set of n points in the plane, not all on a line, contains at least n 2 triples that determine distinct angles (Corr´adi, Erd˝os, Hajnal). − 6. The diameter of any set of n points in the plane with the property that the set of all distances determined by them is separated (on the line) is at least Ω(n). Perhaps it is at least n 1, with equality when the points are collinear. − 7. There is no set of n points everywhere dense in the plane such that all distances determined by them are rational (Erd˝os, Ulam). Shaﬀaf [Sha15] showed that this conjecture would follow from the Bombieri-Lang conjecture.

1.3 COLORING PROBLEMS

If we partition a space into a small number of parts (i.e., we color its points with a small number of colors), at least one of these parts must contain certain “unavoid- able” point conﬁgurations. In the simplest case, the conﬁguration consists of a pair of points at a given distance. The prototype of such a question is the Hadwiger- Nelson problem: What is the minimum number of colors needed for coloring the plane so that no two points at unit distance receive the same color? The answer is known to be between 4 and 7.

GLOSSARY

Chromatic number of a graph: The minimum number of colors, χ(G), needed to color all the vertices of G so that no two vertices of the same color are adjacent. List-chromatic number of a graph: The minimum number k such that for any assignment of a list of k colors to every vertex of the graph, for each vertex it is possible to choose a single color from its list so that no two vertices adjacent to each other receive the same color. Chromatic number of a metric space: The chromatic number of the unit distance graph of the space, i.e., the minimum number of colors needed to color all points of the space so that no two points of the same color are at unit distance.

3 1 7 6 4 1 2 5 6 4 3 2 5 1 7 3 2 4 1 7 5 6 4 1 2 5 6 7 3 2 5 1 7 3 4 1 6 FIGURE 1.3.1 The chromatic number of the plane is (i) at most 7 and (ii) at least 4. (i) (ii)

Polychromatic number of metric space: The minimum number of colors, χ, needed to color all points of the space so that for each color class Ci, there is a distance di such that no two points of Ci are at distance di. A sequence of “forbidden” distances, (d1,...,dχ), is called a type of the coloring. (The same coloring may have several types.) Girth of a graph: The length of the shortest cycle in the graph. A point conﬁguration P is k-Ramsey in d-space if, for any coloring of the points of d-space with k colors, at least one of the color classes contains a congruent copy of P . A point conﬁguration P is Ramsey if, for every k, there exists d(k) such that P is k-Ramsey in d(k)-space. Brick: The vertex set of a right parallelepiped.

FORBIDDEN DISTANCES

Table 1.3.1 contains the best bounds we know for the chromatic numbers of various spaces. All lower bounds can be established by showing that the corresponding unit distance graphs have some ﬁnite subgraphs of large chromatic number [BE51]. d 1 S − (r) denotes the sphere of radius r in d-space, where the distance between two points is the length of the chord connecting them. Next we list several problems and results strongly related to the Hadwiger- Nelson problem (quoted in the introduction to this section). 1. 4-chromatic unit distance graphs of large girth: O’Donnell [O’D00] answered a question of Erd˝os by exhibiting a series of unit distance graphs in the plane with arbitrary large girths and chromatic number 4. 2. Polychromatic number: Stechkin and Woodall [Woo73] showed that the polychromatic number of the plane is between 4 and 6. It is known that for any r [√2 1, 1/√5], there is a coloring of type (1, 1, 1, 1, 1, r) [Soi94]. However, the∈ list-chromatic− number of the unit distance graph of the plane, which is at least as large as its polychromatic number, is inﬁnite [Alo93]. 3. Dense sets realizing no unit distance: The lower (resp. upper) density of an unbounded set in the plane is the lim inf (resp. lim sup) of the ratio of the

TABLE 1.3.1 Estimates for the chromatic numbers of metric spaces.

SPACE LOWER BOUND UPPER BOUND SOURCE Line 2 2 Plane 4 7 Nelson, Isbell Rational points of plane 2 2 [Woo73] 3-space 6 15 [Nec02, Cou02, RT03], Rational points of 3-space 2 2 Benda, Perles 2 1 √3 √3 S (r), r − 3 4 [Sim75] 2 ≤ ≤ 2 2 √3 √3 1 S (r), − r 3 5 Straus 2 ≤ ≤ √3 S2(r),r 1 4 7 [Sim76] ≥ √3 S2 1 4 4 [Sim76] √2 4-space 9 54 [EIL14, RT03] Rational points of 4-space 4 4 Benda, Perles Rational points of 5-space 8 ? [Cib08] d-space (1 + o(1))(1.239)d (3 + o(1))d [FW81, Rai00, LR72] Sd 1(r),r 1 d ? [Lov83] − ≥ 2

Lebesgue measure of its intersection with a disk of radius r around the origin to r2π, as r . If these two numbers coincide, their common value is called the density→of ∞ the set. Let δd denote the maximum density of a planar set, no pair of points of which is at unit distance. Croft [Cro67] and Keleti et al. [KMO16] showed that 0.2293 δ2 0.2588. ≤ ≤ 4. The graph of large distances: Let Gi(P ) denote the graph whose vertex set is a ﬁnite point set P , with two vertices connected by an edge if and only if their distance is one of the i largest distances determined by P . In the plane, χ(G (P )) 3 for every P ; see Borsuk’s problem in the preceding section. It 1 ≤ is also known that for any ﬁnite planar set, Gi(P ) has a vertex with fewer than 3i neighbors [ELV89]. Thus, Gi(P ) has fewer than 3in edges, and its chromatic number is at most 3i. However, if n > ci2 for a suitable constant c> 0, we have χ(G (P )) 7. i ≤

EUCLIDEAN RAMSEY THEORY

According to an old result of Gallai, for any finite d-dimensional point configuration P and for any coloring of d-space with finitely many colors, at least one of the color classes will contain a homothetic copy of P . The corresponding statement is false if, instead of a homothet, we want to find a translate, or even a congruent copy, of P . Nevertheless, for some special configurations, one can establish interesting positive results, provided that we color a sufficiently high-dimensional space with a sufficiently small number of colors. The Hadwiger-Nelson-type results discussed in the preceding subsection can also be regarded as very special cases of this problem, in which P consists of only two points. The field, known as “Euclidean Ramsey theory,” was started by a series of papers by Erd˝os, Graham, Montgomery, Rothschild, Spencer, and Straus [EGM+73, EGM+75a, EGM+75b]. For details, see Chapter 10 of this Handbook.

OPEN PROBLEMS

d 1 1. (Erd˝os, Simmons) Is it true that the chromatic number of S − (r), the sphere of radius r in d-space, is equal to d +1, for every r> 1/2? In particular, does this hold for d = 3 and r =1/√3? 2. (Sachs) What is the minimum number of colors, χ(d), sufficient to color any system of nonoverlapping unit balls in d-space so that no two balls that are tangent to each other receive the same color? Equivalently, what is the maximum chromatic number of a unit distance graph induced by a d-dimensional separated point set? It is easy to see [JR84] that χ(2) = 4, and we also know that 5 χ(3) 9. ≤ ≤ 3. (Ringel) Does there exist any finite upper bound on the number of colors needed to color any system of (possibly overlapping) disks (of not necessarily equal radii) in the plane so that no two disks that are tangent to each other receive the same color, provided that no three disks touch one another at the same point? If such a number exists, it must be at least 5. 4. (Graham) Is it true that any 3-element point set P that does not induce an equilateral triangle is 2-Ramsey in the plane? This is known to be false for equilateral triangles, and correct for right triangles (Shader). Is every 3-element point set P 3-Ramsey in 3-space? The answer is again in the affirmative for right triangles [BT96]. 5. (Bose et al. [BCC+13]) What is the smallest number c(n) such that any set of n lines in the plane, no 3 of which pass through the same point, can be colored by c(n) colors so that no cell of the arrangement is monochromatic? In particular, is it true that c(n) >nε for some ε> 0?

1.4 SOURCES AND RELATED MATERIAL

SURVEYS

These surveys discuss and elaborate many of the results cited above. [PA95, Mat02]: Monographs devoted to combinatorial geometry. [BMP05]: A representative survey of results and open problems in discrete geometry, originally started by the Moser brothers. [Pac93]: A collection of essays covering a large area of discrete and computational geometry, mostly of some combinatorial ﬂavor. [HDK64]: A classical treatise of problems and exercises in combinatorial geometry, complete with solutions. [KW91]: A collection of beautiful open questions in geometry and number theory, together with some partial answers organized into challenging exercises.

[EP95]: A survey full of original problems raised by the “founding father” of combinatorial geometry. [JT95]: A collection of more than two hundred unsolved problems about graph colorings, with an extensive list of references to related results. [Soi09]: A collection of geometric coloring problems with a lot of information about their history. [Gr¨u72]: A monograph containing many results and conjectures on conﬁgurations and arrangements.

RELATED CHAPTERS

Chapter 4: Helly-type theorems and geometric transversals Chapter 5: Pseudoline arrangements Chapter 11: Euclidean Ramsey theory Chapter 13: Geometric discrepancy theory and uniform distribution Chapter 21: Topological methods in discrete geometry Chapter 28: Arrangements

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